INLA Demonstration

Kenneth A. Flagg

Introduction

Why INLA?

  • Bayesian Hierarchical models
    • Many latent Gaussian variables
    • Few parameters
  • Monte Carlo methods impractical
  • Approximate posterior marginals using Laplace expansions
    • Clever algebra, end up with posterior in denominator
    • Taylor expand log-posterior about its mode
    • Results in a Gaussian approximation
    • Some numerical integration needed

Advantages and Disadvantages

  • Advantages
    • Accurate approximation
    • Fast computation of many posterior marginals
  • Disadvantages
    • Does not provide full joint posterior
    • Slow with >4-6 parameters

Derivations

From Blangiardo and Cameletti (2015), Section 4.7

Include derivations of \(\tilde{p}(\boldsymbol{\psi}|\mathbf{y})\), \(\tilde{p}(\theta_{i}|\boldsymbol{\psi},\mathbf{y})\), and \(\tilde{p}(\theta_{i}|\mathbf{y})\); and explanations of mode-finding and CCD.

Illustrations

Simple Normal Example

Blangiardo and Cameletti (2015) section 4.9

  • \(\mathbf{y} = (y_{1}, \dots, y_{n})'\) independent Gaussian observations
  • \(y_{i} \sim \mathsf{N}(\theta, \sigma)\)
  • \(\theta \sim \mathsf{N}(\mu_{0}, \sigma_{0})\)
  • \(\psi = 1/\sigma^{2}\), \(\psi \sim \mathsf{Gamma}(a, b)\)

The posterior distribution of the nuisance parameter is

\[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]

Simple Normal Example

Simple Normal Example

Simple Normal Example

Simple Normal Example

Spatial Point Pattern: Trees in a Rainforest

References

References

Blangiardo, Marta, and Michela Cameletti. 2015. Spatial and Spatio-Temporal Bayesian Models with R - Inla. Wiley.